1 Independent Events Suppose that we flip two fair coins simultaneously on opposite sides of room. Intu- itively, the way one coin lands does not affect the way the other coin lands. The mathe- matical concept that captures this intuition is called independence. In particular, events A and B are independent if and only if:

The milks of all mammals contain lipids but the concentration varies widely between species from c. 2% to greater than 50%(Table 3. 1). The principal function of dietary lipids is to serve as a source of energy for the neonate and the fat content in milk largely reflects the energy requirements of the species, e.g. land animals indigenous to cold environments and marine

4.7.I Occurrence and microheterogeneity B-Lactoglobulin is a major protein in bovine milk, representing about 50% of total whey protein and 12% of the total protein of milk. It was among the first proteins to be crystallized, and since crystallizability was long considered to be a good criterion of homogeneity, B-Ig, which is a typical

Normal bovine milk contains about 3. 5% protein. The conce changes significantly during lactation, especially during the first ft post-partum(Figure 4. 1); the greatest change occurs in the whey fraction(Figure 4.2). The natural function of milk proteins is to supply

Milk is a fluid secreted by the female of all mamalian species, of which there are more than 4000, for the primary function of meeting the complete nutritional requirements of the neonate of the species. In addition, milk serves several physiological functions for the neonate. Most of the non- nutritional functions of milk are served by proteins and peptides which

This is an open-notes exam However, calculators are not allowed You may assume all results from lecture the notes, problem sets and recitation Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem Be neat and write legibly. You will be

YOUR NAME Calculators are not allowed on this exam You may use one 8.5 x 11\sheet with notes in your own handwriting on both side but no other sources of information You may assume all results from lecture, the notes, problem sets, and recitation

Problem 1. An undirected graph G has width w if the vertices can be arranged in a se- quence V1,2,3,…,Vn such that each vertex v; is joined by an edge to at most w preceding vertices. (Vertex vj precedes if i.) Use induction to prove that every graph with width at most w is (w+1)-colorable

Problem set 1 Solutions Due: Monday February 7 at 9 PM Problem 1. The connectives A(and), V(or), and =(implies)come often not only in com uter programs, but also everyday speech. But devices that compute the nand operation

1 Graphs and Trees The following two definitions of a tree are equivalent Definition 1: A tree is an acyclic graph of n vertices that has n-1 edges Definition 2: A tree is a connected graph such that Vu, v E V, there is a unique path connecting u to u. In general, when we want to show the equivalence of two definitions, we must show